Frobenius formula

In mathematics, specifically in representation theory, the Frobenius formula, introduced by G. Frobenius, computes the characters of irreducible representations of the symmetric group S_n. Among the other applications, the formula can be used to derive the hook length formula.

Statement

Let $\chi _{\lambda }$ be the character of an irreducible representation of the symmetric group $S_{n}$ corresponding to a partition $\lambda$ of n: $n=\lambda _{1}+\cdots +\lambda _{k}$ and $\ell _{j}=\lambda _{j}+k-j$. For each partition $\mu$ of n, let $C(\mu )$ denote the conjugacy class in $S_{n}$ corresponding to it (cf. the example below), and let $i_{j}$ denote the number of times j appears in $\mu$ (so $\sum _{j}i_{j}j=n$). Then the Frobenius formula states that the constant value of $\chi _{\lambda }$ on $C(\mu ),$

$\chi _{\lambda }(C(\mu )),$

is the coefficient of the monomial $x_{1}^{\ell _{1}}\dots x_{k}^{\ell _{k}}$ in the homogeneous polynomial in $k$ variables

$\prod _{i<j}^{k}(x_{i}-x_{j})\;\prod _{j}P_{j}(x_{1},\dots ,x_{k})^{i_{j}},$

where $P_{j}(x_{1},\dots ,x_{k})=x_{1}^{j}+\dots +x_{k}^{j}$ is the $j$-th power sum.

Example: Take $n=4$. Let $\lambda :4=2+2=\lambda _{1}+\lambda _{2}$ and hence $k=2$, $\ell _{1}=3$, $\ell _{2}=2$. If $\mu :4=1+1+1+1$ ($i_{1}=4$), which corresponds to the class of the identity element, then $\chi _{\lambda }(C(\mu ))$ is the coefficient of $x_{1}^{3}x_{2}^{2}$ in

$(x_{1}-x_{2})P_{1}(x_{1},x_{2})^{4}=(x_{1}-x_{2})(x_{1}+x_{2})^{4}$

which is 2. Similarly, if $\mu :4=3+1$ (the class of a 3-cycle times an 1-cycle) and $i_{1}=i_{3}=1$, then $\chi _{\lambda }(C(\mu ))$, given by

$(x_{1}-x_{2})P_{1}(x_{1},x_{2})P_{3}(x_{1},x_{2})=(x_{1}-x_{2})(x_{1}+x_{2})(x_{1}^{3}+x_{2}^{3}),$

is −1.

For the identity representation, $k=1$ and $\lambda _{1}=n=\ell _{1}$. The character $\chi _{\lambda }(C(\mu ))$ will be equal to the coefficient of $x_{1}^{n}$ in $\prod _{j}P_{j}(x_{1})^{i_{j}}=\prod _{j}x_{1}^{i_{j}j}=x_{1}^{\sum _{j}i_{j}j}=x_{1}^{n}$, which is 1 for any $\mu$ as expected.

Analogues

In (Ram 1991), Arun Ram gives a q-analog of the Frobenius formula.

See also

• Representation theory of symmetric groups

References

• Ram, Arun (1991). "A Frobenius formula for the characters of the Hecke algebras". Inventiones mathematicae. 106 (1): 461–488.
• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
• Macdonald, I. G. Symmetric functions and Hall polynomials. Second edition. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp. ISBN 0-19-853489-2 MR1354144